Pat the chicken pieces dry with paper towels and season them evenly with salt and black pepper. Keeping the pieces similar in size helps them cook at the same rate.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken in a single layer and cook for 4 minutes without moving, then turn and cook for another 3-4 minutes, until golden and the internal temperature reaches 165°F. Transfer the chicken to a plate.
In a small bowl, whisk together the soy sauce, honey, rice vinegar, sesame oil, garlic, and ginger. In another small bowl, stir the cornstarch and water together until completely smooth.
Pour the soy sauce mixture into the same skillet and bring it to a gentle simmer. Stir in the cornstarch slurry and cook for 1-2 minutes, stirring constantly, until the sauce becomes glossy and thick enough to coat the back of a spoon.
Return the cooked chicken to the skillet and toss well to coat every piece in the sticky sauce. Cook for another minute, allowing the chicken to warm through and absorb the glaze.
Divide the cooked brown rice between four bowls. Add the steamed broccoli, shredded carrots, and cucumber slices around the rice, then spoon the sticky chicken over the top.
Sprinkle each bowl with sesame seeds and sliced green onions. Serve immediately while the chicken and sauce are warm.